The outgoing target and a controlled state class

The baseline belongs to ${\mathfrak{S}}_1$. The class is not a statement about all Hadamard states.

It is nontrivial beyond the baseline. A Weyl displacement by $l$ changes $X(k)$ by a scalar of magnitude $|s(l,k)|\le 2\Vert l{\Vert }_0\Vert k{\Vert }_{\mathrm{U}}$. The displaced Unruh state therefore belongs to ${\mathfrak{S}}_{1+4\Vert l{\Vert }_0^2}$.